Class 13 QED (QD, ED) Queues Erlang-A (M/M/n+G) in the QED & ED Regime

Transcription

1 Service Engineering Class 13 QED (QD, ED) Queues Erlang-A (M/M/n+G) in the QED & ED Regime Motivation, via Data & Infinite-Servers; QED Erlang-A: Garnett s Theorem; The right answer for the wrong reasons - revisited; M/M/n+G: Zeltyn s Approximations (QD, ED); Rules of Thumb; Cost Minimization for Erlang-A (with Zeltyn); Constraint-Satisfaction; The Rule. 1

2 Empirical Service Grade (Beta) QED Erlang-A: Practical Motivation American data. Beta vs ASA beta ASA, sec American data. Beta vs P{Ab} beta % 1% 2% 3% 4% 5% 6% 7% 8% probability to abandon, % 2

3 QED Erlang-A: Theoretical Motivation QED staffing: n R + β R. Assume θ = µ, namely average service-time = average (im)patience. Recall and Note: If θ = µ, the number-in-system of M/M/n+M has the same distribution of a corresponding M/M/ (both are the same Birth&Death process). Formally, in steady-state: L(M/M/n+M) d = L(M/M/ ). The steady-state distribution of M/M/ with parameters λ and µ is Poisson(R), where R = λ/µ (offered-load). For R not too small, Poisson(R) is approximately Normal(R,R). Formally: L(M/M/ ) d R+Z R, where Z is standard normal. We now use these facts to estimate the delay-probability for Erlang- A, in which θ = µ: P{W q (M/M/n+M) > 0} PASTA = P{L(M/M/n+M) n} θ=µ = P{L(M/M/ ) n} Standardizing L R +Z R reveals the QED regime, specifically how square-root staffing yields a non-degenerate delay-probability: P{W q > 0} P Z n R 1 Φ(β). R 3

4 The Erlang-A Queue in the QED -Regime Theorem (with Garnett & Reiman, 2002) The following points of view are equivalent: 0. QED: P{W q > 0} α, for some 0 < α < 1; 1. Manager: n R+β R, for some < β < ; 2. Servers: Occupancy 1 β + γ n ; 3. Customers: P{Ab} γ n, for some 0 < γ < ; in which case α = α(β, µ θ ) = 1 + θ µ h( ˆβ) h( β) 1, which we call the Garnett Delay-Function(s); here ˆβ = β µ θ, and γ = α θ µ [h( ˆβ) ˆβ ]. 4

5 Erlang-A: The Garnett Delay-Functions P {W q > 0} Erlang-A: vs. the QOS P{Wait>0}= parameter vs. β, (N=R+ R) for varying patience θ/µ P{Wait>0} Halfin-Whitt GMR(0.1) GMR(0.5) GMR(1) GMR(2) GMR(5) GMR(10) GMR(20) GMR(50) GMR(100) GMR(x) describes the asymptotic probability of delay as a function of when x. Here, and µ are the abandonment and service rate, respectively. Note: Erlang-C = limit of Erlang-A, as patience indefinitely. 5

6 Understanding the Garnett Functions Erlang-A: P{Wait>0}= vs. (N=R+ R) P{Wait>0} Fix a staffing-level Halfin-Whitt GMR(0.1) (service-grade) GMR(0.5) GMR(1) GMR(2) and let patience : then GMR(5) GMR(10) GMR(20) GMR(50) GMR(100) delays ; in particular, the Garnett functions to the Halfin- GMR(x) describes the asymptotic probability of delay as a function of when Whitt function (infinite-patience). x. Here, and µ are the abandonment and service rate, respectively. Fix a target delay-probability (service level): then, as impatience, less servers (smaller service-grade) are required to achieve the target ( convincing managers to use Erlang-A ). With β = 0 (n = R) and µ = θ, 50% are served immediately. Compare with Erlang-C in which n = R+0.5 R was required. But there is no free lunch: 2% abandon! (under n = 400) see next page. 6

7 Erlang-A: % Abandonment %Ab n vs. β, for varying (im)patience (θ/µ): β GMR(0.1) GMR(0.5) GMR(1) GMR(2) GMR(5) GMR(10) GMR(20) GMR(50) GMR(100) Note the behavior: slope β, for (relatively) large negative β and over all (im)patience levels. For an explanation, think ED: n = R + β R = R γr; hence γ β/ R β/ n, and γ is P {Ab} in the ED-Regime. 7

8 The Right Answer for the Wrong Reason - Revisited If β = 0, the QED staffing level n R + β R becomes n = R = λ µ = λ E[S], which is equivalent to the following deterministic rule: Assign a number of agents that equals the offered load. (Common in stochastic-ignorant operations.) Erlang-C: queue explodes. Erlang-A: Assume µ = θ. Then P{W q = 0} 50%. If n = 100, P{Ab} 4% (twice the value 2% in the graph - why?), and E[W q ] 0.04 E[S] (why?). Overall, reasonable (good?) service level, which will in fact improve with scale. For example, with n = 400, both P {Ab} and E[W q ] reduce to half their value under n = 100 (why?). (Note: Changes in n go hand in hand with same changes in λ, assuming µ remains fixed.) The Effect of Patience: Suppose now µ = 0.1 θ (highly impatient customers). Via the Garnett Functions, suffices n = R R to achieve P{W q = 0} 50%, but this comes at the cost of somewhat over 10% abandoning, with n = 100 (and 5% with n = 400); though E[W q ] decreases to one fourth of the above, assuming µ remains unchanged. 8

9 Erlang-A in the QED Regime: Operational Performance Measures P{W q > 0} 1 + θ µ h( ˆβ) h( β) 1, ˆβ = β µ θ E [W q W q > 0] 1 n 1 θµ [ h( ˆβ) ˆβ ] P{Ab} 1 n θ µ [h( ˆβ) ˆβ ] 1 + θ µ h( ˆβ) h( β) 1 P{Ab W q > 0} 1 n θ µ [ h( ˆβ) ˆβ ] P W q E[S] > t Φ W q > 0 n ( ˆβ + θ µ t ) Φ( ˆβ) P Ab W q E[S] > t n 1 n θ µ θ h ˆβ + t µ ˆβ Here E W q E[S] Φ(x) = 1 Φ(x), Ab 1 1 µ n 2 θ 1 h( ˆβ) h(x) = φ(x)/ Φ(x), hazard rate of N(0, 1). ˆβ ˆβ 9

10 M/M/n+G in the QED Regime agents arrivals λ queue 1 2 abandonment G n µ Density of (im)patience G: g = {g(x), x 0}. Assume g 0 = g(0) > 0. QED regime: n R + β R. QED approximations: Use the Erlang-A formulae (from the previous page), substituting g 0 instead of θ. How to estimate g 0? As θ in Erlang-A! Why? Recall Erlang-A: P{Ab} = θ E[W q ] used for estimating θ (either via θ = [#Abandoning] / [Total Waiting Time]; or by regression of half-hours [%Abandoning] over [Expected-Waits]). M/M/n+G: It turns out that, in the QED regime: P{Ab} g 0 E[W q ]. Hence, one estimates g 0 exactly as θ in Erlang-A. 10

11 Erlang-A: Fitting a Simple Model to a Complex Reality Question: Can one usefully apply the Erlang-A model to systems with non-exponential patience? YES! Erlang-A Formulae vs. Data Averages (Israeli Bank) P{Ab} E[W q ] 250 Probability to abandon (data) Waiting time (data), sec Probability to abandon (Erlang A) Waiting time (Erlang A), sec 11

12 Erlang-A: Fitting a Simple Model to a Complex Reality II 1 P{W q > 0} Probability of wait (data) Probability of wait (Erlang A) Summary: Points: Hourly data (averages) vs. Erlang-A predictions; Formulae with continuous n (special-functions) used to account for non-integer n; Patience estimated via P{Ab}/E[W q ]; Erlang-A estimates provide close upper bounds. 12

13 Fitting Erlang-A Approximations P{Ab} E[W q ] 250 Probability to abandon (data) Waiting time (data), sec Probability to abandon (approximation) Waiting time (approximation), sec 1 P{W q > 0} Probability of wait (data) Probability of wait (approximation) 13

14 Quality-Driven M/M/n+G (QD) Density of patience time at the origin: g 0 > 0. Staffing level: n R (1 + δ), δ > 0. P{W q > 0} decreases exponentially in n. Probability to abandon of delayed customers: P{Ab W q > 0} = 1 n 1 + δ δ g0 µ + o 1 n. Average wait of delayed customers: E[W q W q > 0] = 1 n 1 + δ δ 1 µ + o 1 n. Linear relation between P{Ab} and E[W q ]: P{Ab} E[W q ] g 0 Asymptotic distribution of wait: P W q E(S) > t n W q > 0 e (1 ρ)t, ρ = λ nµ. Comparison with QED: Simpler here, hence worth having. Often, order 1/n replaces 1/ n (though, note conditioning). 14

15 Efficiency-Driven M/M/n+G (ED) Let γ be a QOS parameter, 0 < γ < 1. Assume G(x) = γ has a unique solution x = G 1 (γ), at which g(x ) > 0. Staffing level: n R (1 γ), γ > 0. P{W q > 0} 1. Abandonment-Probability converges to: Offered-Wait converges to x : P{Ab} γ 1 1 ρ. E[V ] x, V p x. Waiting distribution (asymptotically): W q w G, E[W q ] E[min(x, τ)], where G is the distribution of min(x, τ), namely G (x) = G(x), x x ; 1, x > x. 15

16 Operational Regimes: Rules-of-Thumb Assume that the Offered-Load R is not too small (more than several 10 s for QED, more than 100 for ED and QD). ED regime: n R δr, 0.1 δ Essentially all customers are delayed; %Abandoned δ (10-25%); Average-wait 30 seconds - 2 minutes. QD regime: n R + γr, 0.1 γ Essentially no delays. QED regime: n R + β R, 1 β 1. %Delayed between 25% and 75%; %Abandoned is 1-5%; Average wait is one-order less than average service-time (eg. seconds vs. minutes). 16

17 Operational Regimes: Performance Assume that offered load R is not small (more than several 10 s for QED, more than 100 for ED and QD). ED regime: n R δr, 0.1 δ Essentially all customers are delayed; %Abandoned δ (10-25%); Average wait 30 seconds - 2 minutes. QD regime: n R + γr, 0.1 γ Essentially no delays. QED regime: n R + β R, 1 β 1. %Delayed between 25% and 75%; %Abandoned is 1-5%; Average wait is one-order less than average service time (seconds vs. minutes). 17

18 Economies of Scale ( EOS ) For our purpose: Economies of Scale (EOS) prevail if load-increase by a factor m requires staffing-increase by less than m. In what sense Requires? Achieve management goal(s) (constraint satisfaction), or Optimize management goal(s) (optimize cost / profit). Constraint Satisfaction easier to formulate (simpler data) and solve (hence more prevalent); but, as we saw (recall the 80:20 rule), Performance Optimization is easier to grasp. 18

19 Pooling QD Erlang-A s Pool m identical service operations (call centers) with parameters (λ, µ, n, θ). Sustain the same QD operational regime, namely staffing levels: n R + δr, δ = 0.25, for concreteness. Use 4CallCenters to calculate the following: E[S]=6 min, E[τ ]=9 min λ/hr n Occupancy P{Ab} E[W q ] P{W q > 0} % 28.0% 2: % % 10.6% 0: % % 2.5% 0: % % 0.2% 0:01 4.9% 2, % 0.0% 0:00 0.0% 80% 0% 0:00 0% Occupancy converges to 1/(1 + δ); here 1/1.25 = 80%. EOS: Performance Measures improve at an exponential rate. 19

20 Pooling ED Erlang-A s n R γr, γ = 1/6. E[S]=6 min, E[τ ]=9 min λ/hr n Occupancy P{Ab} E[W q ] P{W q > 0} % 38.8% 3: % % 25.2% 2: % % 18.7% 1: % % 16.8% 1: % 3, % 16.7% 1: % 100% 16.7% 1:30 100% P{Ab} and E[W q ] converge as is: P{Ab} γ; E[W q ] γ E[τ]. Thus, in the ED-Regime, there is no EOS for large n. 20

21 Pooling QED Erlang-A s n R + β R, β = 0. E[S]=6 min, E[τ ]=9 min λ/hr n Occupancy P{Ab} E[W q ] P{W q > 0} % 33.6% 3: % % 17.6% 1: % % 8.9% 0: % % 4.5% 0: % 2, % 2.2% 0: % 100% 0% 0: % Delay probability converges to the appropriate Garnett function: P{W q > 0} 1 + θ µ h( ˆβ) h( β) 1 = EOS: P{Ab} and E[W q ] improve at the rate of 1/ n. 21

22 EOS and Constraint Satisfaction Assume service and abandonment rates are as in the previous example: E[S] = 6 min; E[τ] = 9 min. Playing with 4CC yields: ED regime: Loose constraint: P{Ab} 10%. R = 100 n = 91; R = 400 n = 361. Almost no EOS! Use n 90% R (= (1 γ) R, γ P {Ab}). QED regime: Moderate constraint: P{Ab} 2%. R = 100 n = 105; R = 400 n = 399. Saved more than 20 agents: 399 instead of 420 = β = 0.5 for R = 100, β = 0.05 for R = 400. Why EOS? With β fixed, P{Ab} c(β)/ n. Thus, n implies P {Ab}. Consequently, with n, β in order to achieve a given P {Ab} QD regime: Strict constraint: P{Ab} 0.1%. R = 100 n = 119; R = 400 n = 432. More than 45 agents saved: 432 vs = 476. δ = 0.19 for R = 100, δ = 0.08 for R = 400. Why EOS? With δ fixed, P{Ab} decreases exponentially in n, etc. 22

23 Recall: Cost Minimization in Erlang-C (With Borst and Reiman, 2004.) (Equivalently, Profit Maximization, if Revenues proportional to λ.) Cost = c n + d λe[w q ], c cost of staffing; d cost of delay. Erlang-C: Optimal staffing level: n R + β (r) R, r = d/c = delay cost/staffing cost. β (r) = optimal service grade (QOS), independent of λ: β (r) = arg min 0<y< where (recall the Halfin-Whitt function) P w (y) = Very good approximation: 1 + y + r P w(y) y y h( y) 1., β (r) r 1 + r( π/2 1) 2 ln r 2π 1/2 1/2, 0 < r < 10,, r

24 Erlang-A: Staffing via Optimization (with Zeltyn, 2006) We study Minimize Costs (Staffing + Waiting). Why? Comparison easy against Erlang-C; W.L.O.G.: P {Ab} = θ E[W q ] reduces profit- to cost-optimization. Specifically, find n that max. average profit per time-unit: R s λ [1 P n {Ab}] [C s n+c w E n [W q ] λ+c a P n {Ab} λ], where R s is the revenue from a single service. This reduces to c = C s and d = (R s θ + C w + C a θ) in the following: Minimize Cost = c n + d λe[w q ] ; here, as before, c Staffing Cost ; d Delay Cost ; r = d/c. Erlang-A. Optimal staffing level: n R + β (r; s) R, s = µ/θ, β (r; s) = arg min {y + r P w(y; s) s [h(ys) ys]}, y< where (recall the Garnnett functions) P w (y; s) = 1 + h(ys) sh( y) 1. 24

25 Erlang-A: Optimal Service Grade β (QOS) optimal service grade β * µ/θ=0.2 µ/θ=0.4 µ/θ=1 µ/θ=2.5 µ/θ=10 Erlang C waiting cost / staffing cost As θ 0, β (r; µ/θ) increases to β (r) (Erlang-C = M/M/n). r < θ/µ implies that no-service (n = 0) is optimal. Why? d E[τ] < c E[S]: cheaper to let abandon than to serve! r 20 β < 2; r 500 β < 3, as in Erlang-C. Numerical tests exhibit remarkable accuracy & robustness. 25

26 Cost = c N + d λewq (costs: c-staffing, d-delay). Optimal staffing level: * * d µ N R + y ( r; s) R, r =, s =. c θ y * = arg min c y + d ys [ h( ys) ys]. Erlang-A: Actual Cost vs. y Asymptotic Cost < y< P( y ; s) Numerical tests exhibit remarkable accuracy: Actual cost function µ = coincides 1, θ = 1/3with asymptotic cost. Normalized staffing level = ( N R) / R, Normalized staffing level = (n R)/ R; Normalized cost = (Cost cr ) / R. Normalized cost = (cost cr)/ R; P( y ; s) Asymptotic cost: c y + d ys [ h( ys) ys]. Asymptotic cost = c y + d P w (y; s) y s [h(ys) ys], where y = QED service grade

27 Erlang-A: Optimal Staffing λ = 10, µ = optimal staffing level 10 5 pat mean =0:12 (exact) pat mean =0:12 (approximate) pat mean =0:24 (exact) pat mean =0:24 (approximate) staffing level pat mean =1:00 (empirical) pat mean =1:00 (theoretical) pat mean =2:30 (empirical) pat mean =2:30 (theoretical) pat mean =10:00 (empirical) pat mean =10:00 (theoretical) waiting cost / staffing cost waiting cost / staffing cost λ = 100, µ = staffing level pat mean =0:12 (empirical) pat mean =0:12 (theoretical) pat mean =0:24 (empirical) pat mean =0:24 (theoretical) staffing level pat mean =1:00 (empirical) pat mean =1:00 (theoretical) pat mean =2:30 (empirical) pat mean =2:30 (theoretical) pat mean =10:00 (empirical) pat mean =10:00 (theoretical) waiting cost / staffing cost waiting cost / staffing cost 27

28 M/M/n+G: Optimal Staffing Uniformly Distributed Patience. Cost = c n + d λp{ab} optimal staffing level pat mean =0:06 (exact) pat mean =0:06 (approximate) pat mean =0:12 (exact) pat mean =0:12 (approximate) optimal staffing level pat mean =0:30 (exact) pat mean =0:30 (approximate) pat mean =1:15 (exact) pat mean =1:15 (approximate) pat mean =5:00 (exact) pat mean =5:00 (approximate) abandonment cost / staffing cost abandonment cost / staffing cost Cost = c n + d λe[w q ] optimal staffing level pat mean =0:06 (exact) pat mean =0:06 (approximate) pat mean =0:12 (exact) pat mean =0:12 (approximate) optimal staffing level pat mean =0:30 (exact) pat mean =0:30 (approximate) pat mean =1:15 (exact) pat mean =1:15 (approximate) pat mean =5:00 (exact) pat mean =5:00 (approximate) waiting cost / staffing cost waiting cost / staffing cost 28

29 The Rule: Cost Optimization and Constraint Satisfaction Prevalent standard: at least 80% of customers are served within 20 seconds. Call center: λ = 6000/hr, E[S]=4 min (R=400); E[τ]=6 min. 4CallCenters: n = 394 agents required β = 0.3. According to the graph, d/c 1: costs of customers time and servers time are nearly equal. What if d/c = 5? β = 1 n = 420; 82.3% served immediately; 98.9% within 20 seconds. (Comparable Erlang-C: n = 428, corresponding to d/c = 10.) 1.5 θ/µ = 2/3 optimal Quality of Service β * waiting cost / staffing cost 29